document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202701>Question 202701</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following problems for X,Y and Z<BR>\n" );
document.write( "X+Y+2Z=5<BR>\n" );
document.write( "X+Y=5<BR>\n" );
document.write( "X+3Y+Z=11<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152882 by </B><A HREF=/tutors/aboutme.mpl?userid=PRMath><B>PRMath(133)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=PRMath><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=152882\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solve the following problems for X,Y and Z<BR>\n" );
document.write( "X+Y+2Z=5<BR>\n" );
document.write( "X+Y=5<BR>\n" );
document.write( "X+3Y+Z=11\r<BR>\n" );
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document.write( "This is solving a system inthree variables.  So, you should look at your equations above and see that one of the equations has already eliminated a \"Z\" variable.  So, let's choose two equations and eliminate the \"z\".  \r<BR>\n" );
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document.write( "x + y + 2z = 5<BR>\n" );
document.write( "x + 3y + 1z = 11  (To eliminate the \"Z\", let's multiply this equation by -2)\r<BR>\n" );
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document.write( "NEW equations:\r<BR>\n" );
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document.write( "x + y + 2z = 5<BR>\n" );
document.write( "-2x -6y -2z = -22<BR>\n" );
document.write( "_____________________________ (add these two equations)\r<BR>\n" );
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document.write( "-1x -5y = -17  Let's use this equation with the equation above that didn't have a \"z\" variable.\r<BR>\n" );
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document.write( "-1x -5y = -17 <BR>\n" );
document.write( "x + y = 5<BR>\n" );
document.write( "______________(let's add these two equations because I can already see that the<BR>\n" );
document.write( "   -4y = -12  \"X\" variables are going to be eliminated when I add both equations\r<BR>\n" );
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document.write( "Divide both sides by -4<BR>\n" );
document.write( "y = 3\r<BR>\n" );
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document.write( "Now we know that y = 3, so let's \"plug\" that into one of our original equations:\r<BR>\n" );
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document.write( "If y = 3, then we plug in like this:<BR>\n" );
document.write( "x + y = 5<BR>\n" );
document.write( "x + 3 = 5<BR>\n" );
document.write( "x = 5 - 3<BR>\n" );
document.write( "x = 2\r<BR>\n" );
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document.write( "Now we know that y = 3 and x = 2.  Let's plug that info into another equation\r<BR>\n" );
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document.write( "x + y + 2z = 5<BR>\n" );
document.write( "2 + 3 + 2z = 5<BR>\n" );
document.write( "5 + 2z = 5<BR>\n" );
document.write( "2z = 5 - 5<BR>\n" );
document.write( "2z = 0<BR>\n" );
document.write( "z = 0\r<BR>\n" );
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document.write( "Now we have:  x = 2, y = 3 and z = 0. Does this check out? Let's use all our original equations and plug in the numbers.\r<BR>\n" );
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document.write( "X+Y+2Z=5<BR>\n" );
document.write( "2+3+2(0) = 5<BR>\n" );
document.write( "5 = 5 <BR>\n" );
document.write( "YES.  The numbers work for the first equation. Let's check another.\r<BR>\n" );
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document.write( "X+Y=5<BR>\n" );
document.write( "2 + 3 = 5<BR>\n" );
document.write( "5 = 5 <BR>\n" );
document.write( "YES. The numbers work for the 2nd equation.  One more check....\r<BR>\n" );
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document.write( "X+3Y+Z=11<BR>\n" );
document.write( "2 + 3(3)  + 0 = 11<BR>\n" );
document.write( "2 + 9 = 11<BR>\n" );
document.write( "11 = 11<BR>\n" );
document.write( "YES.  The numbers work for the 3rd equation.  We have successfully solved a system in 3 variables.\r<BR>\n" );
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document.write( "For extra info:  \r<BR>\n" );
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document.write( "To solve a system in 3 variables:\r<BR>\n" );
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document.write( "Select ANY two of the 3 equations and work to get one equation in two variables.<BR>\n" );
document.write( "THEN... choose a different pair of equations and eliminate the SAME variable.\r<BR>\n" );
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document.write( "You will then have two equations and can solve for one of the variables.  Use that variable and plug it into the other \"two variable\" equation.\r<BR>\n" );
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document.write( "Now you'll have the answer for two of the variables and.....can easily use an original equation to plug in the two answers to get your third.\r<BR>\n" );
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document.write( "I hope this helps. :-) </SPAN>\n" );
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